{
  "id": "1901.00050",
  "version": "v1",
  "published": "2018-12-31T21:26:01.000Z",
  "updated": "2018-12-31T21:26:01.000Z",
  "title": "Partial smoothness of the numerical radius at matrices whose fields of values are disks",
  "authors": [
    "Adrian S. Lewis",
    "Michael L. Overton"
  ],
  "categories": [
    "math.NA",
    "math.OC"
  ],
  "abstract": "Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of \"disk matrices\" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-31T21:26:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical radius",
      "partial smoothness",
      "disk matrices",
      "arbitrary complex three-by-three matrices",
      "real vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}